Despite belief that Sullivan has been gunning for him, Zola told fans in an email that he felt he had the full support of both owners.
The Times says in an e-mail to the club's supporters, Zola said: "They [Sullivan and Gold] have backed me and I am grateful. I am determined to carry on.
"Losing on Saturday was emotional for everyone but I know we can turn it around and we have the ability to achieve our objectives. My only thought for now is to keep this club in the Premier League and that remains my goal. I'll give everything I can to make this happen between now and the end of the season.
"I have had time to think and we will have to work harder than we have before. I know we can get the results we need and it is just a question of making it happen. The performance was better on Saturday but it is still not the level we should be at. But we can and will do better as a team. I have a great staff and we will work together to find the solutions. We have a responsibility to turn things around and that is what we will do.
"I am here and I am ready to do what I need to do to get the results. There is no doubt about that. We have seen before we are capable of playing at a high level."